Means for describing arcs of large radii



March 7, 1950 w. E. THIBQDEAU 2,499,427

MEANS FOR DESCRIBING ARCS 0F LARGE RADII Filed June 6, 1945 7Sheets-Sheet 1 Fig.1. Fig.2]-

I Wilfred E Thihudeuu LflUZMZWMMM March 7, 1950 w. E. THIBODEAU2,499,427

MEANS FOR DESCRIBING ARCS OF LARGE RADII '7 Sheets-Sheet 2 Filed June 6,1945 QWUW Wil re d Thih u :1 2111.1.

March 7, 1950 w. E. THIBODEAU MEANS FOR DESCRIBING ARCS 0F LARGE RADII'7 Sheets-Sheet 3 Filed June 6, 1945 2192/27/42 glanillll E== Wilfred E.Thibudanu March 7, 1950 w. E. THIBODEAU MEANS FOR DESCRIBING ARCS 0FLARGE RADII '7 Sheets-Sheet 4 Filed June 6, 1945 Wi lfre :1 E. Thibudeuu @54 MQMMM March 7, 1950 w. E. THIBODEAU 2,499,427

MEANS FOR DESCRIBING ARCS 0F LARGE RADII Filed June 6, 1945 7Sheets-Sheet 5 Wilfred E. Thibuc'lanu March 7, 1950 w, THIBQDEAU2,499,427 H MEANS FOR DESCRIBING ARCS 0F LARGE RADII Filed June 6, 1945'7 Sheets-Sheet 6 Wilfred E. Thl'hndeuu Patented Mar. 7, 1950 UNITEDSTATES TENT OFFICE MEANS FOR DESCRIBING ARCS OF LARGE RADII (Grantedunder the act of March 3, 1883, as amended April 30, 1928; 370 0. G.757) The invention described herein may be manufactured and used by orfor the Government for governmental purposes without the payment to meof anyroyalty thereon.

This invention relates to curve-describing and tracing apparatus andmore particularly to instruments intended to describe arcs of circles oflarge radii such that the centers thereof are outside of the area orsurface upon which the arcs are to be drawn or located.

It is a purpose of the invention to provide a relatively simple, compactdevice by which the arcs of circles may be described when the centersthereof are inaccessible or cannot be easily materialized, as forexample, when they are so long as to fall outside the confines of adrawing board or surface or their inclusion in a machine is notconvenient.

Another object is to provide an instrument in which a given are may beduplicated both as to radius and position relatively to points, lines,or surfaces.

A further object is to provide an instrument by which the radius of agiven are may be determined without computation.

Another object is to provide an arc-tracing or arc-scribing instrumentthat may be used in a single operation with simplicity comparable to anordinary compass to define an are that is mathematically exact andaccurate.

A still further object is to provide a method whereby any desired numberof points lying upon the arc of a circle of desired radius may bequickly and expeditiously located without locating the true center ofthe arc.

Another object is to provide a method of laying off a line through agiven point normal to another line or at a given point on said otherline,

Without striking any arcs and using onlylinear measurements. I Otherobjects and advantages of the invention will become obvious as thedescription advances.

In the drawings:

Fig. 1 is a diagram illustrating the geometrical principles upon whichthe invention is based,

Fig. 2 is a diagram showing the geometrical figure by which an involvedvariable of the basic equation upon which my invention is based,.may bedetermined,

Fig. 3 is a simplified diagram of a system of links suitable formaterializing the geometrical features of Fig. 2,

Fig. 4 is a simplified diagram of a system of links as in Fig. 3, towhich have been added other links for materializing the desired locus ofpoints lying upon the arc of a circle, the parts being shown in initialor starting position so that certain parts are superposed,

Fig. 5 is an explanatory diagram showing the system of Fig. 4 with someof the parts rotated through an angle at, I

Fig. 6 is a diagram illustrating the linkage arrangement of another, andat present pre-- yierred, construction, in initial or starting position,

, Fig. '7 is a. diagrammatic view of the linkage of Fig. 6 when moved toan intermediate position during an arc-scribing operation,

Fig. 8 is a schematic plan view of another modi-- fication and showingthe parts in initial position,

. Fig 9 is afview of the modificationof Fig. -8, showing the parts inthe relations assumed dur-l ing the drawing of a curve,

Fig. 10 is a plan view of one form of the invention utilizing theprinciples illustrated at Figs. 4 and 5, and constructed to directlytrace arcs of desired, predetermined radii,

Fig. 1-1 is a view, partly in section, taken upon the linel i-l I, Fig.10,

Fig. 12 is a section taken upon the line l2-l2, Fig. 10, and showing indetail the scriber and on pivot joint of the instrument,

Fig. 13 is a section taken upon the line l3-l3-, Fig. 10, showing theconstruction of the fixed base and the pivotal connections thereto ofthe several links, I

Fig. 14 is a section taken upon the line I l-l4,"v Fig. 10, showing therelation of the parallel links and their connections with the controlmember and right-angled guide,

Fig. 15 is a section taken upon the line I5-I5 of Figs. 11, 12 and 13,and showing the parts rotated to substantially one limiting,arc-scribing" position,

Fig. 16 is a view partly in section, of a modification suitable for usewith the construction of Figs. 10 to 15, inclusive, where it is desiredto vary the length of one of the links of the instrument by varying thedistance between its pivot axes,

Fig. 17 is a section taken substantially upon I the line III1, Fig. 16,and showing the manner of shifting the pivot relatively to the link,

Fig. 18 is a view of a second modified construction for permitting thevariation of the effective i length of one link of the instrument inaccordance with the adjustment provided for in Figs. 16 and 17,

Fig. 19 is a section taken upon the line Iii-49, Fig; 18, i

Fig. 20 is a plan view of a preferred form of the invention showing theparts in initial position,

Fig. 21 is a section taken upon the line 2 l2l of Fig. 20, with theparts in the position prior to the start of an arc-scribing operation.

Fig. 22 is a section taken upon the line 2222 of Fig. 20, and looking inthe direction of the arrows,

Fig. 23 is a side elevation of the instrument taken substantially uponthe line 23-23 of Fig. 20,

Fig. 24 is a plan view similar to Fig. 20, but showing the parts in therelative positions assumed during a curve-scribing operation,

Fig. 25 is a section taken upon the line 2525, of Fig. 20, and showingone sliding connection between one pair of arms of the two right angularmembers,

Fig. 26 is a section taken upon the line 25-26, of Fig. 20, and showingthe other sliding connection between said pair of arms,

Fig. 2'7 is a plan view of another modification of my inventionembodying the principles of operation disclosed at Figs. 8 and 9 andusing rods or tubes for the various linkages,

Fig. 28 is an elevational view of the instrument shown at Fig. 27,looking in the direction of the arrows, Fig. 27,

Fig. 29 is a section taken upon the line 29-29 of Fig. 27 and showingthe pivot axis for the main link as well as the vertical spacing of theseveral links or rods,

Fig. 30 is a section taken upon line 30-30, Fig. 27, showing in detailthe main scriber arm, the scriber slidable thereon and the combinedfixing point and pivot in the base member,

Fig. 31 is a section taken upon line iii-31 of Fig. 27 and showing thesliding pivotal connec= tion between the main link and one arm of theguide member,

Fig. 32 is a view showing the method of constructing an arc of longradius and inaccessible or inconveniently located center using onlystandard drafting instruments,

Fig. 33 shows a convenient chart by which the values of r correspondingto arbitrarily-selected points may be listed,

Fig. 34 is a plan view of an embodiment using,

only three inter-pivoted links for laying off points upon an arc ofpredetermined radius and passing through a selected or desired point,

Fig. 35 is a section taken upon the line 35-45, Fig. 34: showing thepivotal connection between two of the links,

Fig. 36 is a section taken upon the line 36-46 of Fig. 34 and showingthe adjustable pivot connection between two of the links,

Fig. 37 is a side elevation of the end of the main link showing thelatch adapted to be turned down to secure the links in aligned position,and

Fig. 38 is a view showing a chart supplied with each instrument tofacilitate the rapid location of points on are of desired radii.

Referring to Fig. 1, let s be the arc of a circle of radius R and center0, and which it is desired to construct by swinging an arm of initiallength 1'0 about the point p as a center, while continuously increasingthe length of said arm as a function of the angle on so that the distalend of the arm lies upon the arc. Let a represent the instantaneousangle between the initial position m of the scribing arm, 1' thecorresponding instantaneous radius of the scribing arm, and the centralangle between the true radii. The problem initially is, then, to derivean 4 equation in which 1' is expressed as a function Of a.

From the sine law it follows that:

Using the center of the circle as origin and the R line 0:0 as theordinate axis, the equation for the 1 line is:

and the equation for the R line is:

y=x cot 0 Solving simultaneously for ac and y:

R''Tg cot 0cot oz (Rr cot 6 y cot 6cot or gives the coordinates ofintersection of R and r. Since R and r are to intersect along the arc ofthe circle:

1: t R2 Therefore:

'0) 2 (cot 0-cot a) 080 0= (cot 0-cot a) cot 0= A cos a-cos a+A cosraw-Jim cos win m cos oz+1/A sin a It should be noted also, that whereEquation 2 may be solved geometrically as shown at Fig. 2 wherein theangle a is laid off between a base D and a side of length equal to oneunit to any convenient selected scale, and the third side is constructedof length A to the same scale. Then the length of an altitude dropped tothe base is, evidently, sin a, and the length of the base is cos a+ /Asin 0:. Comparing these values with Equation 2 it will be noted that theratio of r to m is exactly determined by the ratio of the sum of thesides of the triangle of Fig. 2, to its base. For convenience, let thevalue of cos e+ /A sin cc be represented by D throughout the balance ofthe description.

Fig. 3 illustrates in simple diagrammatic form, a suitable assembly ofparts for materializing and simulating to scale, the triangle of Fig. 2.B00 is a unitary right-angular member, rotatable about the center 0 andwhose initial position is B'OC'. A link of unit length, to the scale ofthe instrument, is fixed along the direction 0C" and has one endcoinciding with the center 0'. A second link, of length A has one endpivoted to the other end of the unit link, and its remaining endpivotally and slidably connected with the arm C. Now, as member BOG isrotated through successive values of the angle on, the distance fromcenter 0 to the position of the end of link A along arm 0, is thecorrect value of D for the corresponding value of 0:. Thus, the linkageshown may be used as an elementary and component part of an instrumentin which the value of r from center 0, along arm B may be determined foreach value of the angle a, or may, in itself comprise an instrument fromwhich the value of r may be computed.

Referring to Fig. 4, let BOC represent the uni tary right-angularmember, as in Fig. 3. Two straight links M and E are shown superposed inthis figure, link M being pivoted on arm C at a fixed distance A+1 fromcenter 0, and slidable along and over arm B. Link E is pivoted on arm Bat the adjustably fixed distance 10 from point 0, and is pivotally andslidably connected with the outer end of link A so that the intersectionof link E with arm C, is maintained at the distance D from center 0.This distance is, of course, unique for each corresponding value of 0:.If it be assumed that links M and E are maintained parallel, then, asmember BOC is rotated about center 0 and one end of link A pivots aboutthe adjacent end of the fixed unit link while its other end slides alongarm C, similar triangles are maintained such that the point ofintersection of link M with arm B is the value of r for thecorresponding Value of or. In short, as member BOC is pivoted, asaforesaid, the point materialized by the intersection of arm B and linkM traces the arc of a circle of radius R.

Fig. shows the linkage of Fig. 4 rotated through an angle or, whereincrossed links a and b are used to maintain links M and E at all 6 timesparallel. From the construction previously described, it will be notedthat so that the locus of points formed by the intersection of link Mwith arm B is the desired arc of a circle of radius B.

At Figs. 10 to 15, inclusive, I have shown a form of instrumentembodying the principles of my invention and based upon the diagrammaticillustrations of Figs. 4 and 5. In these figures, the numeral Iidentifies a base having spaced apertures within which pointed fixingscrews 8 and 9, Fig. 13, are threaded as an aid in maintaining the linejoining the axes of said screws in coincidence with the chord of the arcto be drawn. A pivot pin I0 fits a bore in base I in such a positionthat the distance between the axes of pin I0 and screw 8 represents oneunit to any convenient scale. Since ratios only are involved in theoperation of this portion of the instrument, the said unit distance mayhave any practical value that will be determined by the physical sizeand capacity of the model being constructed.

A right-angled guide member II includes a hub I2 bored to fit pin III,and two parallel arms I3 and I4 extending radially from said hub andbore and spaced axially of pin Ill, as clearly shown at Fig. 12. In theinterests of rigidity and ease of operation, arms I3 and I4 areconnected at their outer ends by a spacing collar I5. A specialsupporting bolt I6 passes through collar I5 and aligned holes in arms I3and I4 to rigidly connect said arms in parallel relation, by means of anut I'I. Bolt I6 has its lower end shaped to revolubly mount and retaina ball I8 and is so dimensioned that, when ball I8 rests upon a planesurface, arms I3 and I4 will be substantially parallel thereto. It willbe noted, too, from Figs. 10, 12 and 15, that arms I3 and M are providedwith slots I9 and 20, respectively.

Guide member II also has a portion shown in cross section upon Fig. 13,and comprising an arm 2I, slotted as at 23 and extending from hub I2 atright angles to arms I3 and I4. At its outer end, arm 2I carries athickened base part 22 so dimensioned that its lower surface is coplanarwith the lower surface of base I when the parts are assembled in normaloperating condition. It will thus be, appreciated that guide member IImay pivotas a unit about the axis of pin III, relatively to base I. Thedistance between screw 8 and pin I0 is fixed and represents unitdimension to any convenient scale which dimension in thepresent exampleremains fixed in direction during an arc-scribing operation.

A link 24 has a slot 25 therein extending for practically its entirelength. As shown at Fig. 12, means are provided whereby arm I3 of memberII and link 24, may be pivotally secured together at various selecteddistances from pin I0, depending upon the radius of the arc to betraced.

' This means may consist of a pivot joint including a bolt 21 fitting acentral bore in a bushing 28 that has portions of difierent diameters,separated by a flange. The larger diameter portion fits between thesides of slot 25. The flange. which may be square, has a transversedimension to fit between shoulders formed by rabbeting the corners ofslot 24 on the side adjacent link 28,

the dimensions being such that the flanged sur-' face extends slightly-afew thousandths of an inch, above the adjacent surface of the link.

Furthermore, the larger diameter portion of bushing 28 terminates shortof the lower surface of link 24. The smaller diameter portion of bushing28 extends through slot I9 in arm i3 and slightly beyond the adjacentend of a second bushing 29 rotatably fitting the smaller diameterportion of bushing 28 and having an outer diameter to fit slot I9.Bushing 29 has a flanged head at one end having a dimension slidablyfitting between shoulders formed by rabbeting the corners of slot E9 onthe side adjacent link 24. A nut 30 is threaded upon the projecting endof bushing 29. A washer 3| is interposed between nut 39 and arm I3, andhas a projection shown at Figs. 10, 11 and 12 which acts as an indicatorin cooperation with a scale on the side of arm I3 to give the setting ofbolt 2'! in terms of the true radius of the arc to be drawn. A secondnut 2? is threaded upon bolt 27 and, when turned down, engages theprojecting smaller diameter portion of bushing 28, to clamp link 2 1between the flange on said bushing and the head upon bolt 2?. Also whennut is turned down, it acts to clamp arm I3 between the shoulder onbushing 29 and washer 3|. Since all pivotal movement between the linksand arm appears as a relative rotation between bushings 28 and 29, thesebushings may, if desired, be square on exterior cross section on thoseportions fitting their respective slots. Thus,

the pivot axis may be locked relatively to both link 24 and arm I3, bysimply tightening both nuts 3!! and 21'. Or, by leaving one nut loosenedthe said axis may have sliding pivotal association with thecorresponding link or arm. Thus, for example, when nut 3|) is turneddown, while nut 21 is loosened, the pivot axis of bolt 21 is fixedrelatively to arm I3 while, at the same time, link 24 may pivot andslide thereon.

The lower end of link 2Q extends beneath the slot in arm 2|. A bolt 32projects through both arms and carries aligned sleeves 33 having aslightly greater total axial dimension than the combined thickness oflink 22 and arm 2| and an outer diameter snugly fitting the slot 23 inarm 2| and an aperture in link 24. A nut 3Q is threaded onto the end ofbolt 32 to hold the arm and link against separation while, because ofthe aforesaid dimension of sleeves 33, the parts 2| and 24 may freelypivot about and slide relatively to bolt 32. In short, bolt 32materializes a point determined by the intersection of the longitudinalaxes of arm 2| and link 2 From Fig. 13, it will be noted that the headedend of bolt 32 has a pintle concentric of its axis and fitting theaperture in one end of a link 36. The other end of said link is pivotedon screw 3, there being a sleeve 33 on the screw, in-- terposed betweenbase i and link 35 to maintain the latter in the proper plane ofoperation. The effective length of link 35, will be recognized as thedistance A in the schematic illustration of Fig. 5 whereby, as guidemember I I is swung about the axis of pin Ill, link 33 is also pivotallymoved about screw 8 and a sliding movement of bolt 32 along slot 23, isefiected to thereby vary the aforesaid point of intersection of arm 2|and link 24 along arm 2 I.

A link 33 has a slot 39 extending from one end for a substantialdistance. The other end of the link is pivoted to the thickened basepart of arm 2|, as by a bolt 42 and it will be noted that the distancebetween the axes of pin I0 and bolt 40, materializes the dimension (A+1)of explanatory Fig. 4. Incidentally, it should also be noted that, inFigs. 11, 13 and 14, the axes of bolts 32 and 40 are shown in alignmentmerely because the parts are there shown in the initial or startingposition and that said axes move out of alignment as soon as thearc-tracing operation begins, as will be evident from inspection of Fig.15.

Link 38 has an integral oiTset portion 4| (see Figure 15) provided witha slot 42 and is connected to be maintained in parallelism with link 24,by means of pivotally connected crossed links it and 42 having rollerson their ends fitting the respective slots in a manner obvious frominspection of Figs. 10 and 15. The intersection of link 38 with arm I ldefines the tracing point of the desired arc. This point is materializedby sleeves 35 fitting slots 39 and 23. A scribing device 46 has a shankjournaled within sleeves 35 and a tubular extension adapted to receive alead or tracing point 4'! urged outwardly by a spring 48, Fig. 12. Thedevice 46 may be provided with a split collar 49 and a clamping screw 55to hold the lead 41 in adjusted position when desired. Arm it isprovided with a gooseneck portion 5|, slotted as at 52, to accommodatesleeve 31 .as arm I i approaches its position shown on Fig. 15.

The operation of the species just described, will now be clear from theforegoing description. The base 7! is positioned so that the points 8and 9 lie upon the chord of the are desired to be drawn with the axis ofpin iii coincident with What will be the mid ordinate of said arc. Nut38 is then loosened and bolts 32 and 49 being concentric, links 24 and38 are pivoted as a unit about the presently common axis of these boltsto thereby move bolt 27 and tracer 4? until their presently common axislies on the mid point of the arc to be drawn. Nut 30 is now tightened tofix the pivot bolt 2'! at the distance it from the axis of pin ID.

Guide member II is now rotated about pin ill whereupon, the axis of bolt48 traverses an arc of radius (A+1) about the axis of pin It. Because oflink 36, bolt 32 is constrained to slide along slot 23 and thusdetermine the continuously varying location of bolt 32 (i. e. the postof arm 24) along arm 2| for each corresponding value of the angle ofrotation of member H from initial position, i. e., the angle a ofEquation 2. As another point on link 2% is determined by bolt 2! and nut30 at the set distance m from pin It measured along arm I3, the positionof link 24 is determined and unique for each value of the angle a.Likewise the position of link 38 is determined since it is pivoted atone end by bolt in, to member I I, while, at the same time, it ismaintained. parallel to link 24. Thus the element 4'! materializing theintersection of link 38 with arm I4, describes the desired arc inaccordance with Equation 2.

The radius of the arc can be varied by adjustment of either T0 or A. Achange in To alone will give arcs of varying heights and correspondingvarying radius of are due to the constancy of R R7'Q Changes in A, onthe other hand, To being constant will yield a series of arcs throughthe extremity of re. Thus, variability of both A and T0 will affordextreme flexibility as to the radius of the arc and its height.

In Figs. 16 and 1'7 I have shown, a construction that may be embodied inthe species of Figs. 10 to 15, inclusive, when it is desired to vary thevalue of A. Since the distance between pin I3 and bolt 9 in is (A+1),the effective value or A may be conveniently varied by changing thedistance between parts It! and 40. In the construction shown, 1'represents a base to which the arms I3 and I4 of member l2 are pivotedby means of pm H). Member I2 is provided with a cavity 53 havingparallel Side walls and within which slides a block 54, bored to receivepivot pin In which extends upwardly from base 1' into said block througha slot 55 in member l2. A plate 55 is secured to member I2 over the openside of cavity 53 and has a threaded opening and a scale 51 concentricto said opening.

A thumbscrew 58 engages the threaded opening and is swiveled to block54, as at 59 so that, when turned, block 54 is translated relative tocavity 53 and the effective distance between pin I8 and bolt 49 isvaried. The head Bl] of screw 58 has a pointer El coacting withgraduations to gage the amount of turning of the screw and to repeat anydesired previous setting of the instrument. The gage may be graduated interms of R/ro. It is of interest to note the small adjustment requiredfor the distance A for correspondingly large changes in radii. Forexample, if R/ro=10, A=1.111, while if R/T0=20, A=l.053. Thus a changeof .058 unit in the eiiective length of the arm 2| will double the valueof R for any selected value of m. It will be noted too, that this, aswell as the modifications subsequently to be described, are capable ofdescribing the complete arc, since the linkages representing unitdistance and the distance A+l, will operate for their intended purposewhether the upright carrying the scriber is swung through the first orthe second quadrant.

Figs. 18 and 19 show a form of adjustable pivot for accommodating andcorrespondingly adjusting the effective length of link 36 whenadjustment is made in the effective value of A by turning of knob 69,Figs. 16 and 17. In Figs. 18 and 19, 8' is the pivot screw fixed in base1'. A threaded sleeve 62 has a flange at one end fitting beneath thehead of screw 8. A stud 63 has a head fitting within sleeve 62 andhaving its shank projecting through a central opening in a cap 64threaded upon sleeve 62. The end of stud 63 projects through a slot 65in arm 36 and is clamped in adjusted position along the slot by a nut 68threaded upon stud 83. When adjustment in the effective value of A is tobe made, nut 66 is loosened, adjustment of knob 50 to the desired valueas indicated upon scale 57 is made and nut 65 again tightened. It willbe understood that the axes of pivot pin l0, fixing screw 8, and bolts32 and 40 are coplanar at the time this adjustment is made.

Figs. 6 and 7 show a second form of the invention wherein O is theorigin or mid-point of the chord of an arc to be drawn. A link of unitlength extends from O to the left and is fixed to the drawing surface. Amember comprising integral, angularly-related arms B and C, is pivotedat point 0, at the juncture of said arms. A link G is pivotallyconnected at one end with arm C, at a distance (A+1) from point 0 sothat said link intersects arm B. A second member has angularly relatedarms, one of which coincides with, and is slidably connected with, arm Cso that its other arm F is maintained at all times parallel to arm B.Arms B and F coincide upon Fig. 6. The second member projects to theleft of point 0 and there has its end pivotally connected with one endof a link of length A. The other end of this link is pivoted to the endof the unit link remote from point 0. Link G is pivoted to arm F at adistance T0 measured from arm C along arm F. Thus, as the two aforesaidmembers are pivoted about 0, arm F is slid along arm C so that itsdistance from the pivot between arm C and link G is at all times equalto the distance D, as defined in connection with Fig. 2. From inspectionof Fig. 7, it is clear that 1/1o=(A+1) /D so that, in accordance withthe basic equation, the point of intersection of link G with arm B,determines the locus of points of the desired arc.

Figs. 20 and 26, inclusive, depict the form of the inventionmaterializing the arrangement of Figs. 6 and '7. In these figures 19identifies a base that may conveniently be in the form of a rectangularbar having pointed screws H and 12 threaded into apertures adjacent itsends whereby the base may be fixed in line with the chord of an arc tobe drawn. A member 13, generally in the shape of a T, has a head 14slotted at one end as at 15, and an arm or shank l6, slotted as at H andprovided at its free end with a support 18 whereby the member issupported parallel to the surface upon which the instrument is beingused. A pivot pin 19 is secured within a bore in base 18, and head 14 ispivoted on this pin at 80, at the point of intersection of thelongitudinal axes of head 14 and shank 16.

A right-angled member 8| has arms 82 and 83. Arm 82 has a longitudinalslot 84 extending from one end to a point about midway of the arm. Pin19 extends through this slot and the arm is held against the head of pin19 by means of a spacing collar 85. The other end of arm 82 has aprojection 85 of the same axial dimension as collar 85 and a headedscrew 81 passes through slot 15 in head 14 and engages a threadedaperture in said projection. In this manner arm 82 is mounted forsliding movement only relatively to and along head 14 so that arms 16and 83 are maintained parallel at all times.

A link 88, of effective length A, is pivoted at one end to base 19, bymeans of a screw 89, and at the other end to arm 82, by means of a screw99. From Figs. 20, 24 and 26, it will be noted that head 14 has a slot9| loosely fitting around screw 89 so that the parts 18, 14 and 82 ma bemoved into the initial superposed position shown in Fig. 20. A link 92has parallel undercut longitudinally-extending channels in its upper andlower sides, as indicated at and 96, Fig. 22. Link 92 is pivoted at oneend to head 14, by means of a screw 94. A pivot bolt 93 has a headshaped to slidably fit its channel 95 (Fig. 22) and the smaller portionof this head has a diameter greater than the width of the slot 98 and,furthermore, projects a little above the upper surface of link 92. A nut100 engages the shank of bolt 93 and, when turned down, clamps bolt 93in position relatively to arm 83, while, because of the constructionpreviously described, the head of said bolt may at the same time slidealong channel 95 in link 92. A washer 99, between nut I90 and arm 83 mayhave lugs fitting slot 98 to prevent turning relatively thereto. Oneedge of this washer acts as an indicator over a scale l8! formed alongone edge of arm 83. This scale may be graduated in terms of radii of thearcs desired to be scribed.

A pivot bolt I92, Fig. 22, has a head snugly fitting channel 98 with itsshank extending through slot 11 in arm 16. A sleeve 193 of the sameouter diameter as the width of slot 11, surrounds said shank. A flangedscriber barrel I84 engages the threaded end of bolt I02 projecting belowslot TI. As the sleeve I03 projects a little below the lower surface ofarm 16, the flange n barrel I04 has a slight clearance with arm 16 sothat the scriber barrel and bolt I02 may slide freely along both arm andlink 92. A scriber I05 slidably fits an axial bore in barrel I94 and isurged outwardly by a spring I06. It will be noted that channel 95 needbe only relatively short, while, to provide maximum capacity for a giveninstrument, chan nel 96 should extend to the outer end of its link 92.

The operation will be clear from the previous description including thatadvanced in connection with Figs. 6 and '1. The instrument with parts inthe relative position shown in Fig. 20, is located so that the points ofscrews H and 12 are on the chord of the arc to be scribed, with the axisof pin 19 at the center of the desired chord. Nut )9 is loosened andlink 92 is swung about pivot screw 94 until the indicator edge of washer99 indicates the desired radius on scale IIlI The nut is then tightenedto fix bolt 93 relatively to arm 83. Shank 16 is then caused to pivotabout pin 19. Since link 92 is fixed at one end to head 14 and at aremote point to arm 83 of member 8!, while the screw 90 in the other arm82 of said M member is constrained by 88 to circular motion about screw89, pivoting of shank 16 eifects sliding of arm 83 relatively thereto.Thus scriber I05 is caused to travel outwardly along arm It as theintersection of said arm with link 92 moves outwardly in proportion toturning of member 13. As explained in connection with Figs. 6 and 7,this intersection at all times lies on the locus of points of a circulararc.

In Figs. 8 and 9, I have shown diagrammatically another construction bywhich the basic principles of the invention may be carried out. Arms Band C, that are preferably but not necessarily mutually normal, arepivoted about their intersection at point 0. A link H is pivoted to armC at a distance from 0 equal to 70(A+1) for any value of To, althoughTo(A+1) need not be actually measured. A slide is mounted for movementalong arm C and carries an upright of unit length, with its outer endpivotally connected with link H, and spaced from the pivot point of linkH and arm C by the initial distance (A-I-l). The basic triangle (to theleft of 0) comprises the fixed side, of unit length, the side of lengthA pivoted to the unit side, and the third side D, connected at one endto point 0 and at the other end to the aforesaid slide, carrying theupright.

As link H and arms B and C are conjointly rotated about 0, the uprightis shifted so that its distance from the intersection of link H with armC is maintained at D by the basic deformable triangle. The device thuscontinuously sets up similar triangles such that, with To equal to theinitial intercept of link I-I upon arm B,

and the intercept of H and B has a distance from O at all times equal tothe desired radius (or polar coordinate of the arc) In Figs. 27 to 31,inclusive, there is shown a modification embodying the principles ofoperation disclosed at Figs. 8 and 9 and in which the variousgeometrical figures are simulated by means of rods or tubes. A base IEOhas apertures spaced a unit distance, center to center. Each of theseapertures receives a threaded pivot I51 and IE2. Each pivot is pointedat its lower end and is held in relation to base I68 by nuts threadedthereon and engaging opposite sides of the base as shown at I63, Figs.28 and 30. Each pivot has a shank of reduced diameter at its upper end,as shown at I64. The end of a rod I65, of effective length A, is forkedat one end, as at I65, and pivoted upon the reduced shank of pivot I61.A pivot block I61 has a bore receiving the other end of rod I65 and isadapted to be secured in adjusted position along said rod, by means of aset screw I68. A second block I68 is pivoted to block I61, as at I10, onan axis that is normal to the axis of rod I65. Block I69 has an aperturereceiving a rod I1! and adapted to be fixed in adjusted position on saidrod, by means of a set screw I12. Thus, by loosening screws I68 and I12,blocks I51 and I69 may be adjusted as a unit, to thereby vary theeffective value of the dimension A.

A pivot block I13 has an upper bore slidably receiving rod I11, and aparallel lower bore within which fits the end of a base rod ISII. Theend of rod I88 within this bore is forked, as clearly shown at Figs. 28and 30, to receive the adjacent end of link I15. The reduced end ofpivot I62 passes through aligned apertures in block I13, the forked endof rod I80, and link I15, whereby the link ma pivot relatively to theblock and rod, while the block and rod are united in fixed relation. Arod or arm I15 is secured to block I13 and extends perpendicularly torods HI and I811. The outer end of rod I18 carries a support comprisinga block I11, adjustably fixed along the rod by means of a set screw I18,and having a leg I19 of the length necessary to support rod I16 parallelto the drawing surface.

At its other end, rod I has a block I8I slidably mounted thereon andsecured in adjusted position by a set screw I82. A support I83 isthreaded into the lower side of this block and a pivot bolt Iil isthreaded into the top side thereof. A hub 185 is journaled on bolt I84and has a bore radial of the axis of said bolt accommodating a rod I86.As the bolt also passes through a transverse hole in the rod, the huband rod are thereby fixed together. A member I81 has a bore slidablyfitting base rod I89. This member also has a second parallel bore withinwhich fits the slotted end of rod I11. This slot receives one end of alink I88 and a pivot pin I89 passes through aligned holes in the memberI81, the forked end of rod HI, and link I88, to pivotally connect linkI88 and member I81, while fixedly uniting the rod and member. Thus rodsIll and I80 are mounted for pivoting in unison about the axis of pivotI62, while, at the same time, rod I1I may be reciprocated axially alongrod I89 so that the axes of the two are at all times parallel. The otherend of link I88 is pivoted between the forked ends of rod I86, as shownat I90, Fig. 28.

A second hub I 9| is journaled upon bolt I84 immediately above hub I andhas a radial bore accommodating a rod or link I 92 that passes over andintersects arm I16 throughout the range of movement of the instrument.As bolt I84 passes through a transverse hole in link I92, the latter isthereby fixedly united with hub I9I.

Member I81 has an arm I93 of unit effective length, secured thereto bythreading, as shown at Fig. 31. The outer end of this arm is pivotallyconnected with a sleeve 1%, by means of a pivot screw I85. Link I92slidably fits sleeve I94. The

intersection of arm I15 and link I92, is materialized by sleeves I96 andI91, pivotally connected by a screw I98 intersecting the bore of bothsleeves at right angles. A scriber or stylus barrel I99 is threaded intoan aperture in the lower side of sleeve I96 and carries a stylus orscriber 260. A spring I is positioned within the barrel and acts to urgescriber 20D outwardly into engagement with the drawing surface. It willbe noted that the connection afiorded by link I88 prevents member I81from any rotation or wobble about the axis of base member I80 while theinstrument is being adjusted for the selected value of 1.

In operation, with the parts in initial position, as shown inFigs. 27and 28, set screws I12 and I 82 are loosened and slide I96 is movedalong scale 202 on arm I16 to the desired value of To, while block I8Icorrespondingly moves along base rod I80. As rod I86 is maintained inalignment with base rod I80 at this time, and is connected with rod l'IIthrough link I88, rod III is correspondingly axially translated. Setscrews I12 and I82 are now tightened and the instrument is located withscriber 206 at the mid-point of the are it is desired to draw and withthe point of pivot I62 on the radius of the are through said point whilethe pivot point of I6I lies upon the chord of the arc through pivot I62.Hi6, and link I92 are swung about the point of pivot I62 as a center.Since the side ofthe basic triangle of unit length, materialized by thedistance between the points of pivots IBI and I62 on base I60, remainsfixed, rod I65 pivots about the axis of pivot I6I as the parts arerotated, and rod I?! is axially translated relatively to rod I86 inaccordance with the change in the dimension D, which varies as afunction of the angle on. This dimension is transmitted by rod In tovary the distance between pivot pin I69 and bolt I84. As this distancevaries, arm I93 is translated along rod I86 to inversely vary theangular relation between link I92 and base rod I80 so that theintersection of link I92 with arm I16, moves outwardly along said arm asthe parts are rotated. Thus scriber 20!] traverses and describes thedesired arc.

It will be understood that, whereas, in the embodiment shown, theadjustment for the values of A is indicated as controlled by, a setscrew, a finer adjustment may be provided in various ways andconstruction well known in the art.

Arm I16, rods HI and e In Fig. 32 the basic triangle has sides A and U.

A semicircle of radius A is described about the end of U and radiallines drawn from the other end of U (point 0) at various values of a.

If A+U be taken equal to 1'0 we have TD: (11+ U) From the values of thisequation at a=180 it will be clear that Therefore, when it is desired todraw an arc of a certain radius, through a given mid-point P of saidarc, a line is first drawn through said point representing a portion ofthe radius to said point from the inaccessible or inconveniently-locatedSubstituting thevalue A=4-U in Equation 3 and solving, we get U=1.'765",and A=2.235". Now the distance 2.235" is laid off from P along theaforesaid radius and the point 0, representing one end of the unitlength is thus located. The distance 1.765" is then laid ofi from thispoint along the radius, to locate the other end 0' of said unit lengthor the point from which values of 'r are to be measured.

An arc of radius 2.235" is struck, using C as a center and covering thedesired angular extent about point 0, of the arc to be drawn. This areis divided into convenient points that may be selected at random such asthe points I, 2, 3 etc. and radial lines of convenient lengths are drawnthrough these points and point 0. From the fundamental equation,T=To(A+U) /D, and since by construction, we have made To and (A+U) equalto 4", the value of 1' measured along any radial line through point 0and the respective points I, 2, 3, etc., will be i /D or 16/D, where Dis the distance from O to the selected point I, 2, 3, etc. ,A chart asin Fig. 33 is formed and values of D are taken from Fig. 32 with anengineer's scale, and listed in the appropriate column. A slide rule isnow used to compute the values of 1' as listed in the chart and thesevalues of r are laid ofi along the respective radial lines from pointO.v A fair curve is then drawn through the points thus located to givean arc of the desired radius through the desired point. The entireprocedure may be rapidly performed with only one setting of the slidingscale of the slide rule and gives a mathematically exact arc of anydesired radius. It is interesting to note that, when A=U, R becomesinfinite and the computations give a straight line.

This suggests that the method disclosed is of great utility where it isdesired to locate two lines at right angles to each other with anextremely high degree of accuracy, as in laying out building or boundarylines or in tool design or manufacture. Note that, in contradistinctionto the 3-4-6 triangle method commonly used, the present method requiresno intersecting arcs but only the laying out of distances and that theaccuracy is limited only by the limits of accuracy in reading a tape,scale or micrometer. While for convenience, To and (A-l-U) have beentaken as equal, this is by no means necessary since in a given problem,To, A and U are constants and the ratio of A+U to 1'0 may have anyproportionality factor Ic'other than unity.

At Fig. 34 I have shOWn an embodiment of the .invention well adapted tofacilitate the location of any desired number of points upon an arc ofdesired radius through a given point. From Equation 3 it will be notedthat, for a fixed value of U, the value of A for any fixed radius of arcis fixed. For example, suppose U is given a value of 2". Then, for anarc of radius of 19", A must have a value of 2.65; and A+U=4.65. Thisprinciple is conveniently utilized in the instrument of Figs. 34 to 38,inclusive, where the numeral 2IIl indicated a unit link shown as 2" inlength between the axes of a pivot screw 2H and a second pivot screw2I2. Both pivot screws are identical and 2!! is shown at Fig. 35.

It will be noted that the screws have pointed ends for penetration ofthe supporting surface ing shorter arm 2 l5 approximately 6 in lengthand slotted as shown at 219. Arm 2M is graduated in inches and tenths,as shown at 21% with the axis of screw 2!! as origin. A slider 2H ismounted upon arm 2 l4 and consists of a simple frame embracing the edgesof the arm 2M. If desired, the slide may have inwardly-directed sideflanges engaging longitudinal grooves in the sides of the arm, or suchflanges may be omitted so that the slide can be lifted off the arm. Theframe carries. a window 2 it having mutually normal indicator linesscored therein with a hole at the intersection of said lines wherebypoints may be located lying exactly upon the radial line from the ofscrew 2| l.

The arm 215. of main link 2L3, is slotted, as at 2H and a scale 22B isprovided therealong, and so graduated that the distance of pivot bolt22l from the axis of screw 2i i, is indicated at the reading edge of aslide or washer 222 positioned on said pin and maintained in the angularposition shown by a lug fitting between the side edges or the slot andlocated at said reading edge. A clip 23?! is pivoted on lugs at the freeend of arm 255. This clip has projections 23l which, when said clip isturned 90 from the position shown at Fig. 37, roject below the lowersurface of arm 255 and may thus engage the sides of A link 224, to holdit inexact alignment with arm 2 I 5.

Link 22 t is slotted as at 225 and both slots 2 l9 and 225, arerabbetted as shown at 225 to provide shoulders. The pivot, includingbolt 22I passing through both slots H9 and 225, is of the sameconstruction as that shown at 36, Fig. 12, so that it is sufiicient topoint out, in connection with Figs. 34 and 36, the sleeves 221 and 232,the nut 22:; threaded upon sleeve 232 and the nut 229 threaded upon bolt22L Thus, when both nuts are loose, the pivot bolt may be slid freelyalong slots 2 l G- and 225, while when both nuts are tightened, the link22% is locked to arm 2i 5. With nut 229 tightened, and 228 loosened, thebolt is locked to link 224, but may slide relatively to arm 2 I 5.

To facilitate its use, I propose to provide a chart to accompany eachinstrument, as shown at Fig. 38, and having a first column in which aregiven the radii of the arcs, a second column giving corresponding valuesof A for the constant U of the instrument (2 in the model shown), athird column with values of A+U and a fourth column giving correspondingvalues of (A+U) To locate points upon the arc of a circle ofpredetermined radius and passing through a predetermined point, thevalue of (A-l-U) and (A-l-U) for that radius, are taken from the chart.The links and arms are aligned by the use of clip 238 and pivot bolt 22!is slid until the reading edge of slide 222 indicates the value of A-l-Uon scale 221i. Nut 22$ is then tightened to lock bolt 22! to link 22a.The point of screw 2! l is then located at the point through which it isdesired to have the arc pass, while the point of screw 2i 2 ispositioned upon a radius of the circle in case it is desired to scribethe are about a center located in a definite direction. Link 243 is nowrotated about screw 2! I through any desired angle. This movement causeslink 224 to pivot and slide 222 to move along arm 2 i 5. The indicated,value on scale 22s is then read and usedv to divide into thepreviously-determined valuev of (A+U) as by means. of a slide rule.Then, without moving link 213, slide 2i i is moved until the value ofthe quotient just determined, is indicated by the transverse line onwindow 2K8. A mark is then made through the hole in 16 said window. Thelink H3 is pivoted to a new position and another point located alongscale 2l6. In this way, as many points as desired may be located uponthe arc.

The foregoing procedure is but one of the many Ways in which theinstrument may be used. In fact, by applying the Equations 2 and 3,.arcs of circles of any length of radius whatsoever, may be precisely andaccurately determined, it being noted that, in the model shown, ro=A+ U,where U equals the unit distance of the basic equation.

It will be understood that a device similar to that of Fig. 34 may beconstructed in which the U and A links have fixed lengths n and 11Arespectively, 1' being calculated from Equation 2 for the various valuesof D as determined by the base of the n, nA, D triangle.

It is within the purview of my invention to substitute a chart for Fig.38, wherein the relations between R, A+ U, and (A-l-U) are given by twocurves having values of R as abscissae and values of A-l-U and (A+U) asordinates. Such curves will obviate the necessity for interpolation inthose cases Where the desired radius is not an integer.

While I have shown members ll, 14, as. having their constituent parts orarms in mutually normal relation it will be clear that this is forconvenience only, and that said parts may within practical limits haveangular relations other than While I have shown a preferred form of myinvention, together with several modifications, numerous changes,substitutions of equivalents and rearrangement of parts will occur, orbe obvious to those skilled in this art. Therefore, I do not wish to belimited to the precise details of construction shown. On the contrary, Iwish to reserve all such changes, modifications, arrangements andsubstitutions as fall within the scope of the subjoined claims.

For instance, while several forms have been shown whereby the finalresult for r is reflected through similar triangles after mechanicalsolution of D it will be obvious that many such arrangements arepossible and will be obvious in various combinations. and on the variouscoordinate ax'es'or inintermediate axes. And while the unit or U side.of the basic triangle has been shown as the fixed. side, it may be theside movable through the angle a. While graduations have been shown,these may be omitted and the distances scaled as in the use of anordinary compass. While the fixed points of the basic triangle has beenshown as coinciding with the pivots thereon, such is not. essential andthis sizemay be fixed at any point and have any convenient length.

Throughout the specification the device has been referred toas one. fordescribing an arc. Thus it is not intended that it be limited to anyinstrument for scribing an arc. Conceivably it might be employed forcarrying a grinding or cutting tool, space considerations being of primeimportance in machinery.

Alternatively, Equation 3 may be expressed in the form Thus, when it isdesired to first assume a value inconveniently-positioned center, thecorresponding value of U may be obtained, often by mere mentalcomputation. As an example, suppose it is desired to lay out an arc of acircle of 20 feet radius and passing through a given point P and thatfeet is selected as a convenient length for A. Then and (A+U) =64. Theconsecutive distances A and U are then laid off from P along a known orselected radius of the arc to locate points such as C and 0, Fig. 32. Asuitable number of arbitrarily-selected points corresponding to thepoints I, 2, 3, 4, etc., Fig. 32, are located upon the circumference ofa circle having point C as a center.

Now, taking a tape and locating its zero graduation at point 0, thedistance to point I is measured, divided into 64, and without moving thetape, the quotient is laid off, to locate a first point of the desiredarc. Points 2, 3, etc., are similarly treated and corresponding pointsupon the desired are are located.

It should be noted that, while the method has been described as beingcarried out with plotting centers located upon the concave side of thearc, that is, between the arc and its true center, this is by no meansnecessary. In fact, one highly useful aspect of the invention is thatarcs of any radius may be determined and generated from the convex side.Such a procedure might be desirable, for example, where an arc must bedrawn closely adjacent and with its concave side facing, the edge of adrawing, or where an arcuate wall, side wall, or embankment is to beconstructed along the shore of a bay or bank of a river. In such casesit is merely necessary to remember that the radius R is negative withrespect to the values of A and U and should be so written in the use offormula (3).

To take the example previously used, suppose it is desired to lay outthe arc of a circle of 20 radius from the convex side thereof, and thata value of A of 4 is chosen. Formula (3) is now applied in which thevalue R=20 is used. Thus the value of U=6' is obtained and (A+U) =100.The point P is then selected through which the arc is to pass, andsecond and third points are located along the radius, extended, at 4'and 10' from P. With the second point as a center, any desired number ofmeasuring points upon the circumference of a circle of radius 4 areselected so as to embrace the angular extent of the arc to be drawn. Thedistances of these points from the aforesaid third or A-l-U point arethen measured, and convenientl tabulated and each distance so obtainedis divided into 100. The resulting values are then laid off along theradial lines through the respective measuring points to locate pointsupon the desired arc and through which a fair curve may be passed. Thusthe process is carried out with equal facility irrespective of the sideof the arc upon which the plotting and measuring points are located. Itwill be noted that, in all cases where the plotting and measuring pointsare upon the concave side of the arc to be determined, A has a numericalvalue greater. than u, while, when the aforesaid pointsv are to belocated upon the convex side of the arc,.A has a numerical value lessthan u. When the two are equal, the resulting arc is a straight line',aspreviously'explained. While I havcde 1'8 scribed the process as that ofplotting or describing an are it will be realized that it is operablethroughout 360 and hence may be used to form a complete circle whendesired.

In this way, any number of points may be located upon the arc of acircle having any length of radius whatsoever and using only a limitedspace. Furthermore since only linear measurements are involved, thepossibilit of error is reduced to a minimum, While the accuracy andprecision of the method are limited only by the instrument used tomeasure and lay off distances. I

In general, this invention is basic to the triangle linkage which solvesthe complex portion of Equation 2, and it should be noted that a basictriangle having any values whatever for its two collapsing sides willdetermine a value of D for some circular arc. Likewise, whereadjustments for the value of A have been shown in some modificationssuch may be included in any form of the invention.

I claim:

1. In an arc defining instrument, a base having first and second pivotaxes, first and second members each comprising first and secondangularly-related arms, said first arms being connected for coincidentalsliding movement to maintain said second arms parallel, said firstmember being pivoted on said first pivot axis, a link pivoted on thefirst arm of said first member at a point remote from said first pivotaxis and also pivoted on the second arm of said second member, said linkintersecting the second arm of said first member, and means constraininga point on the first arm of said second member to rotate about saidsecond pivot as a center.

2. An arc scribing instrument comprising a base having first and secondparallel pivot axes spaced in a first direction, a first member havingfirst and second angularly-related arms extending radially of said firstpivot, a second member shaped to define first and Second angularlyrelated arms, means connecting said first arms for coincidental slidingmovement, a'

link pivoted at predetermined points on said first arm of said firstmember and said second arm of said second member, means constraining apoint on the first arm of said second member to circular movement aboutsaid second pivot axis, and scriber means controlled jointly by saidlink and said second arm of said first member, to materialize the pointof intersection thereof as said members are swung about said first pivotaxis.

3. In an instrument for scribing arcs of long radii, a base having meansdefining a first pivot axis having means whereby to be fixed to asurface, first and second members having first arms extending radiallyof said axis and connected for parallel relative sliding movement only,said first nember being pivoted on said axis, said members having secondarms extending in parallelism angularly from said first arms, meansdefining a second pivot axis spaced unit distance in one direction fromsaid first axis, a link of predetermined efiective length pivoted at onepoint on said second axis and at a second point on said first arm ofsaid second member, a second link pivoted on the first arm of said firstmember at a distance from said first pivot axis equal to the sum of saidpredetermined length and unit distance in the other direction, meanspivotally connecting said second link at a predetermined variabledistance along the second arm of said sec- 19 ond member,.and a scriber-jointly controlled by said second link and said second arm of.saidfirst member to continuously materialize their point ofintersection as said members are .conjointly rotated about said firstpivot axis.

4. In an are drawing instrument, a first memher having angularly-relatedfirst and second arms extending radially .from a first fixed 'pivotaxis, a second member slidable in a pathalong said first arm, a linkpivoted at one point to said first arm and at a second point to saidsecond member and offset from said path, means constraining a point onsaid second member, lying in said path, to rotation in a fixed pathabout a second fixed pivot axis, and a scriber jointly controlled bysaid link and second arm continuously materializing their point ofintersection.

5. A curve-tracing instrument comprising a first member rotatable abouta first pivot axis and defining angularly-related first and second pathsradially of said axis, a second-,member constrained to sliding movementonly, along said first path, a first link pivoted on said first-memberat one point in said first path and at another point on said secondmember ,ofiset from said first path, means constraining a point on saidsecond member lying in said first path to rotation about a second pivotaxis spaced from said first pivot axis, and scriber means slidable alongsaid first link to materialize its intersection with said second path.

6. In a curve-scribing instrument. a first member rotatable about afirst pivot axis, said member defining angularly-related first andsecond paths extending radially from said first axis, a second memberconnected for translation on said first member along said first path, alink pivoted on said first member at a point in said first path apredetermined distance on one side of said first pivot axis, meanspivoting said link on said second member ata point offset to one side ofsaid first path, means guiding a point on said second member in acircular path of predetermined radius about a second pivot axis spacedunit distance from said first axis, said point on said sec- ,ond memberlying on said first path on the other side of said first pivot axis, andscriber means controlled by said link to -materialize itsintersection-with said second path.

'7. In an arc-scribing instrument, a first member pivoted about a firstpivot axis and defining first and second mutually normal guide pathsextending radially of said axis, a secondmember connected for pivotalmovement with said first member and for translation relatively theretoalong said first path, a first link pivoted on said first member at apoint on one side of said first pivot axis lylngin said first path,means pivotally connecting said first link and second member at apointoffset an adjustabledistance from said first path, a second link pivotedat one end to said second member at a point in said first path on theother side of said first-pivot axis and at its other end to a secondpivot axis spaced iromsaid first pivotaxisand scriber means slida ablealong said first link to automatically materialize the-pointofintersection of said first link and second guide path.

8. In a scribing instrument, a pair of members .slidably connected vforrelative translation along a first path extending radially of a pivotaxis common to said members, a link pivoted on one member on an axisnormal to andintersecting said path, there being a longitudinal undercutchannel on each .side of said link, a first pivot bolt "having a jihea-dslida-bly guided "in cone said channel and slidably fitting ;a slot inthe other membensaid slot defining apath normal :to said first path, asecond pivot bolthaving a head-slidably guided in the other channel andslidably fitting a slot in said one member, and scriber means carried bysaid first pivot bolt.

9. In a curve-scriber, first and second members each havingright-angularly related first and second arms, said first armsbeingconnected for relative sliding movement in the directionof their lengthto maintain said second arms parallel, said first member being pivotedon afirst fixed point at the intersection of its arms,:a link pivoted onsaid first arm of said first member and said second arm of said secondmember, and means constraining a pointon the first arm of said secondmember to rotation about a'second fixed point on a predetermined radius,and means controlled by said link and second arm of said first member,to scribe the locus of their intersectionas said members are rotatedabout said first point.

'10. A device for describing an arc of a circle comprising arms fixed atright angles to each other, afirst lever articulated on'one of the arms,a second lever articulated at the intersectionof the two arms, a thirdlever articulated to the second lever, an arm articulated to the distalend of the third lever and arranged to slide along the arm to which-thefirst lever is articulated and carrying an arm slidingly engaging thefirst lever to turn'it about its point of articulation in response tomotion of the sliding arm.

11. A device as in claim 10 wherein said second and third levers bear alength ratio of Rm to 'R where R is theradius of the desired arc and Tois the predetermined ,maximumheight of the are above ,its chord.

12. A device .as inclaim 10 :wherein said second and third levers bear alength ratio of R-To to R Where R is the radius of the desired arcand Tois the predetermined maximumfheight of the are above its chord and saidarm carried by the sliding arm .has a vertical height equal to To.

13. A device as in claim 10 wherein said second and third levers bear alength ratio of Rr0 to R where R is the radius of the desired arc androis the predetermined maximum height of the arc above its chordand saidarm carried by the sliding arm has a vertical'height equal to unity.

14. A device for mechanically solving the value of the expression cos a+/A sin 0: comprising an arm, .a first lever articulated to the arm, ,asecond lever articulated to the first lever and ar .ticulated to andslidable along the arm, the ratio of the efiective lengths of the'leversbeing 71.: 14A and a being the angle between the first lever and the armas the second lever slides along the arm.

15. ,In aninstrumentjor describing the arcs of .circles, a membercomprising first and second angularlyerelated rods extending. radiallyof and rotatable as a .unit about afirst axis, first and second blocksslidable on said first ,rod, means fixing said first block in adjustedposition along its rod,.a third rod parallel to and slidably connectedwithsaid first rod through said second block, a link pivoted on saidfirst block and intersecting said second rod, anarm extending from saidsecond blockand having pivotal sliding connection with said link,interpivoted slidable, sleeves on said second rod and link respectively,to materialize th ir intersection, and .means 10- 21 tatable about asecond axis spaced a predetermined distance from said first axis, andconnected to slide said third rod and thereby said second block and arm,as said rods are rotated as a unit about said first axis.

WILFRED E. THIBODEAU.

REFERENCES CITED The following references are of record in the file ofthis patent:

France Apr. 14, 1922

